Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 032, 13 pages      arXiv:0711.4550
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Equivariance, Variational Principles, and the Feynman Integral

George Svetlichny
Departamento de Matemática, Pontifícia Unversidade Católica, Rio de Janeiro, Brazil

Received November 02, 2007, in final form March 13, 2008; Published online March 19, 2008

We argue that the variational calculus leading to Euler's equations and Noether's theorem can be replaced by equivariance and invariance conditions avoiding the action integral. We also speculate about the origin of Lagrangian theories in physics and their connection to Feynman's integral.

Key words: Lagrangians; calculus of variations; Euler's equations; Noether's theorem; equivariance; Feynman's integral.

pdf (217 kb)   ps (165 kb)   tex (17 kb)


  1. Saunders D.J., The geometry of jet bundles, Cambridge University Press, 1989.
  2. Olver P.J., Equivalence, invariants and symmetry, Cambridge University Press, 1995.
  3. Svetlichny G., Feynman's integral is about mutually unbiased bases, arXiv:0708.3079.
  4. Kolar I., Michor P.W., Slovak J., Natural operations in differential geometry, Springer, New York, 1993, available at
  5. Olver P.J., Applications of Lie groups to differential equations, Springer, New York, 1986.
  6. Otterson P., Svetlichny G., On derivative-dependent infinitesimal deformations of differentiable maps, J. Differential Equations 36 (1980), 270-294.
  7. Svetlichny G., Why Lagrangians?, in Proceedings XXVI Workshop on Geometrical Methods in Physics (July 1-7, 2007, Bialowieza, Poland), AIP Conference Proceedings, Vol. 956, Editors P. Kielanowski, A. Odzijewicz, M. Schlichenmeier and T. Voronov, AIP, New York, 2007, 120-125.
  8. Bengtsson I., Three ways to look at mutually unbiased bases, quant-ph/0610216.

Previous article   Next article   Contents of Volume 4 (2008)